Effective bending modulus of carbon nanotubes with rippling deformation

This paper concerned with the relation of the bending moment to the bending curvature during bending of carbon nanotubes, and the relation between the rippling formation and the bending modulus. Based on the three-dimensional orthotropic theory of finite elasticity deformation, a non-linear bending moment–curvature relationship of carbon nanotubes which is the appearance of wavelike distortion on the inner arc of the bent nanotubes is simulated by using an advanced finite element analysis package, ABAQUS. Utilizing the non-linear bending moment–curvature relationship, the effective bending modulus of carbon nanotubes with different cross-sections are obtained by means of a bi-linear theory and a simplified vibration analysis method. The effective bending modulus of carbon nanotubes simulated in the paper is close to the measuring result presented in reference Science 283 (1999) 1513].